December  2019, 24(12): 6817-6835. doi: 10.3934/dcdsb.2019168

Flocking of Cucker-Smale model with intrinsic dynamics

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, China

* Corresponding author: Xiaoping Xue

Received  September 2018 Revised  February 2019 Published  July 2019

Fund Project: The authors are supported by NSF of China grants 11731010 and 11671109

In this paper, we consider the flocking problem of modified continu- ous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents' own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant $ L $ of the intrinsic dynamics and exponent $ \beta $ measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large $ L $ or unconditionally for $ \beta>0 $. At last, we provide several numerical simulations to illustrate our theoretical results.

Citation: Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168
References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

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S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

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H. BaeY.-P. ChoiS.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

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Y. CaoW. RenD. Casbeer and C. Schumacher, Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.  doi: 10.1109/TAC.2015.2479926.  Google Scholar

[6]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

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F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

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F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

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F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.  doi: 10.1137/100785910.  Google Scholar

[14]

F. Dalmao and E. Mordecki, Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.  doi: 10.1109/TAC.2012.2188440.  Google Scholar

[15]

J.-G. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.   Google Scholar

[16]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

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[21]

C.-H. Li and S.-Y. Yang, A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[22]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[23]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[24]

Z. LiS.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[25]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[28]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[29]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[30]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[31]

L. RuZ. Li and X. Xue, Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.  doi: 10.1016/j.jfranklin.2014.12.007.  Google Scholar

[32]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. Google Scholar

[34]

C. SomarakisE. ParaskevasS. Baras and N. Motee, Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.  doi: 10.1109/TCNS.2017.2765824.  Google Scholar

[35]

H. SuG. ChenX. Wang and Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.  doi: 10.1016/j.automatica.2010.10.050.  Google Scholar

[36]

Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp. doi: 10.1063/1.4929496.  Google Scholar

[37]

Y. SunY. Wang and D. Zhao, Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.  doi: 10.1016/j.physa.2015.08.005.  Google Scholar

[38]

T. V. TonN. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.  Google Scholar

[39]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

T. VicsekA. Czir$\acute{o}$kE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[41]

T. Vicsek and A. Zefeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[42]

W. YuG. Chen and M. Cao, Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.  doi: 10.1109/TAC.2011.2112477.  Google Scholar

[43]

W. YuG. ChenM. Cao and J. Kurths, Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.   Google Scholar

show all references

References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[3]

H. BaeY.-P. ChoiS.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[4]

S. Bolouki and R. Malham$\acute{e}$, Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2012. doi: 10.1109/Allerton.2012.6483385.  Google Scholar

[5]

Y. CaoW. RenD. Casbeer and C. Schumacher, Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.  doi: 10.1109/TAC.2015.2479926.  Google Scholar

[6]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[7]

I. D. CouzinJ. KrauseN. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[8]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[9]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[13]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.  doi: 10.1137/100785910.  Google Scholar

[14]

F. Dalmao and E. Mordecki, Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.  doi: 10.1109/TAC.2012.2188440.  Google Scholar

[15]

J.-G. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.   Google Scholar

[16]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

[17]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[20]

A. JadbabaieJ. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[21]

C.-H. Li and S.-Y. Yang, A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[22]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[23]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[24]

Z. LiS.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[25]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[28]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[29]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[30]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[31]

L. RuZ. Li and X. Xue, Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.  doi: 10.1016/j.jfranklin.2014.12.007.  Google Scholar

[32]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. Google Scholar

[34]

C. SomarakisE. ParaskevasS. Baras and N. Motee, Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.  doi: 10.1109/TCNS.2017.2765824.  Google Scholar

[35]

H. SuG. ChenX. Wang and Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.  doi: 10.1016/j.automatica.2010.10.050.  Google Scholar

[36]

Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp. doi: 10.1063/1.4929496.  Google Scholar

[37]

Y. SunY. Wang and D. Zhao, Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.  doi: 10.1016/j.physa.2015.08.005.  Google Scholar

[38]

T. V. TonN. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.  Google Scholar

[39]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

T. VicsekA. Czir$\acute{o}$kE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[41]

T. Vicsek and A. Zefeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[42]

W. YuG. Chen and M. Cao, Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.  doi: 10.1109/TAC.2011.2112477.  Google Scholar

[43]

W. YuG. ChenM. Cao and J. Kurths, Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.   Google Scholar

Figure 1.  Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 3] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 3] $
Figure 2.  Evolutions of $ d_V(t) $ and $ d_X(t) $ with 6 particles: {(a)} $ d_V(t) $ for $ t\in [0, \; 3] $; {(b)} $ d_X(t) $ for $ t\in [0, \; 3] $
Figure 3.  Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 150] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 150] $
Figure 4.  Evolutions of $ d_V[t] $ and $ d_X[t] $ with 6 particles: {(a)} $ d_V[t] $ for $ t\in [0, \; 150] $; {(b)} $ d_X[t] $ for $ t\in [0, \; 150] $
Figure 5.  Dynamics of velocity $ V_2 $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
Figure 6.  Evolutions of $ d_V[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
Figure 7.  Evolutions of $ d_X[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
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